Topological field theory and the quantum double of SU(2)

TL;DR

This paper explores the quantum double structure of SU(2) in topological field theory, revealing new insights into particle interactions, braid statistics, and scattering in 2+1 dimensions, with implications for quantum gravity.

Contribution

It introduces the quantum double D(SU(2)) in the context of topological particles coupled to Chern-Simons theory, and analyzes its representations and physical implications.

Findings

Quantum double D(SU(2)) describes particle symmetries in 2+1D topological systems.

Derived fusion rules and braid statistics from the quantum double structure.

Computed differential cross sections for nonabelian Aharonov-Bohm scattering.

Abstract

We study the quantum mechanics of a system of topologically interacting particles in 2+1 dimensions, which is described by coupling the particles to a Chern-Simons gauge field of an inhomogeneous group. Analysis of the phase space shows that for the particular case of ISO(3) Chern-Simons theory the underlying symmetry is that of the quantum double D(SU(2)), based on the homogeneous part of the gauge group. This in contrast to the usual q-deformed gauge group itself, which occurs in the case of a homogeneous gauge group. Subsequently, we describe the structure of the quantum double of a continuous group and the classification of its unitary irreducible representations. The comultiplication and the R-element of the quantum double allow for a natural description of the fusion properties and the nonabelian braid statistics of the particles. These typically manifest themselves in generalised Aharonov-Bohm scattering processes, for which we compute the differential cross sections. Finally, we briefly describe the structure of D(SO(2,1)), the underlying quantum double symmetry of (2+1)-dimensional quantum gravity.